mlsanityData to 13 Oct 2026

The price of ignorance: how much does it cost to forget noise structure in low-rank matrix estimation?

arXiv 2205.10009 ↗First posted 20 May 2022cs.IT cs.LG math.ITcited by 9

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Abstract (arXiv)

We consider the problem of estimating a rank-1 signal corrupted by structured rotationally invariant noise, and address the following question: how well do inference algorithms perform when the noise statistics is unknown and hence Gaussian noise is assumed? While the matched Bayes-optimal setting with unstructured noise is well understood, the analysis of this mismatched problem is only at its premises. In this paper, we make a step towards understanding the effect of the strong source of mismatch which is the noise statistics. Our main technical contribution is the rigorous analysis of a Bayes estimator and of an approximate message passing (AMP) algorithm, both of which incorrectly assume a Gaussian setup. The first result exploits the theory of spherical integrals and of low-rank matrix perturbations; the idea behind the second one is to design and analyze an artificial AMP which, by taking advantage of the flexibility in the denoisers, is able to "correct" the mismatch. Armed with these sharp asymptotic characterizations, we unveil a rich and often unexpected phenomenology. For example, despite AMP is in principle designed to efficiently compute the Bayes estimator, the former is outperformed by the latter in terms of mean-square error. We show that this performance gap is due to an incorrect estimation of the signal norm. In fact, when the SNR is large enough, the overlaps of the AMP and the Bayes estimator coincide, and they even match those of optimal estimators taking into account the structure of the noise.

Key facts. Cited by 9 papers in the map (data through 13 Oct 2026). Builds on 27 of its 93 references that are in the map. First posted on arXiv 20 May 2022.

Builds on

27 of 93 references.

References · 27

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15 more
  1. 2014Phase transitions and sample complexity in Bayes-optimal matrix factorizationcited by 22
  2. 2020Approximate Message Passing with Spectral Initialization for Generalized Linear Modelscited by 22
  3. 2017The Computer Science and Physics of Community Detection: Landscapes, Phase Transitions, and Hardnesscited by 20
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  5. 2021Approximate Message Passing for orthogonally invariant ensembles: Multivariate non-linearities and spectral initializationcited by 18
  6. 2021Estimation in Rotationally Invariant Generalized Linear Models via Approximate Message Passingcited by 18
  7. 2019Inference with Deep Generative Priors in High Dimensionscited by 17
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  11. 2020Empirical Bayes PCA in high dimensionscited by 11
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Cited by year

Citing papers per year: 2022: 2; 2023: 2; 2024: 3; 2025: 2.

9 papers.

  1. 2024Matrix Denoising with Doubly Heteroscedastic Noise: Fundamental Limits and Optimal Spectral Methodscited by 5
  2. 2024On the phase diagram of extensive-rank symmetric matrix denoising beyond rotational invariancecited by 4
  3. 2022Bayes-optimal limits in structured PCA, and how to reach themcited by 3
  4. 2022Approximate Message Passing for Multi-Layer Estimation in Rotationally Invariant Modelscited by 3
  5. 2023Gradient flow on extensive-rank positive semi-definite matrix denoisingcited by 1
  6. 2024Unrolled denoising networks provably learn optimal Bayesian inferencecited by 1
  7. 2025Sparse corruption in low-rank matrix inference: the PCA benchmarkcited by 1
  8. 2023Mismatched estimation of non-symmetric rank-one matrices corrupted by structured noisecited by 0
  9. 2025Computational and statistical lower bounds for low-rank estimation under general inhomogeneous noisecited by 0